RLC Circuit Analysis

Differential Equations · Axiom Academy

The same second-order equation that governs a shock absorber also governs a radio tuner — inductor, resistor, and capacitor standing in for mass, damping, and spring. You're tuning a radio receiver's RLC circuit — an inductor L , resistor R , and capacitor C in one loop — and every design choice you make bends the SAME differential equation three engineers before you have already bent for springs, shocks, and doors. Charge the capacitor, cut the source — watch it ring With no driving voltage, — the exact analog of a released spring-mass system. Drag the resistance and watch the stored charge either overshoot-and-settle, or just crawl home. Now drive it — find the station An AC source adds to the loop. Solve for the steady-state current and its amplitude peaks exactly when the driving frequency matches the circuit's own natural frequency — that's the station locking in. , , — swap those three and a spring-mass system and an RLC circuit are the SAME differential equation wearing different units. Drag the damping and watch both wobble (or not) in lockstep. One second-order equation, A y'' + By' + Cy = f(t) , keeps showing up wearing different labels: mass-spring-damper , RLC circuit , even a building swaying in an earthquake or a chemical concentration settling to equilibrium. Learn the roots-and-discriminant playbook once — overdamped, critical, underdamped, resonance — and it transfers everywhere the labels change but the equation doesn't.

This is the written version of the interactive lesson above. See the full Differential Equations course.